What is the general solution for -3=cot^2(θ)+3csc(θ) ?

The general solution for -3=cot^2(θ)+3csc(θ) is θ=(3pi)/2+2pin,θ=(7pi)/6+2pin,θ=(11pi)/6+2pin

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-3=cot^2(θ)+3csc(θ)

- 3 = cot^{2} (θ) + 3 csc (θ)

θ = \frac{3 π}{2} + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Solution steps

- 3 = cot^{2} (θ) + 3 csc (θ)

Switch sides

cot^{2} (θ) + 3 csc (θ) = - 3

Subtract

- 3

from both sides

cot^{2} (θ) + 3 csc (θ) + 3 = 0

Rewrite using trig identities

2 + csc^{2} (θ) + 3 csc (θ) = 0

Solve by substitution

csc (θ) = - 1, csc (θ) = - 2

csc (θ) = - 1 : θ = \frac{3 π}{2} + 2 πn

csc (θ) = - 2 : θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Combine all the solutions

θ = \frac{3 π}{2} + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

arccos (e^{- x}) = \frac{π}{3}

cos (θ) = - \frac{11}{170}

2 cos^{2} (θ) - 7 cos (θ) + 3 = 0, 0 \leq θ \leq 2 π

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) = 1

sin (x) cos^{2} (x) = sin (x)

What is the general solution for -3=cot^2(θ)+3csc(θ) ?
The general solution for -3=cot^2(θ)+3csc(θ) is θ=(3pi)/2+2pin,θ=(7pi)/6+2pin,θ=(11pi)/6+2pin